2024/03/07 by Purnaprajna Bangere, Bangere, Purnaprajna, Francisco Javier Gallego +3
Computer Science · Mathematics · #14B05 #14B10 #14D06 #14D15 #14D20 #14E30 #14J10 #14J45 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.2403.04167
openalex publication_date 2024/03/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this article, we introduce a new approach to show the existence and smoothing of simple normal crossing varieties in a given projective space. Our approach relates the above to the existence of nowhere reduced schemes called ribbons and their smoothings via deformation theory of morphisms. As a consequence, we prove results on the existence and smoothing of snc subvarieties V ⊂ ℙN, with two irreducible components, each of which are Fano varieties of dimension n>2, embedded inside ℙN for effective values of N, by the complete linear series of a line bundle H. The general fibers of the resulting one parameter families are either smooth Fano, Calabi-Yau or varieties of general type, depending on the positivity of the canonical divisor of their intersections. An interesting consequence of projective smoothing is that it automatically gives a smoothing of the semi-log-canonical (slc) pair (V, Δ), where Δ= cH, c < 1, is a rational multiple of a general hyperplane section of H. For threefolds, we are able to give explicit descriptions of the smoothable snc subvarieties due to the classification results of Iskovskikh-Mori-Mukai. In particular, we show the existence of unions V = Y1 \bigcupD Y2 ⊂ ℙN, where Yi's are smooth anticanonically (resp. bi-anticanonically) embedded Fano threefolds, intersecting along D, where D is either a del-Pezzo surface or a K3 surface (resp. a smooth surface with ample canonical bundle) and their smoothing in ℙN to smooth Fano or Calabi-Yau threefolds (resp. to threefolds with ample canonical bundle) for various values of N between 10 and 163. In cases when the general fiber is a smooth Fano or Calabi-Yau threefold, one can choose c such that (V, Δ) is a Calabi-Yau pair while in all cases c can be chosen so that (V, Δ) is a stable pair.